\(\int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx\) [376]
Optimal result
Integrand size = 24, antiderivative size = 192 \[
\int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {35 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{128 \sqrt {2} a^{5/2} d}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}+\frac {7 i \cos (c+d x)}{48 a d (a+i a \tan (c+d x))^{3/2}}+\frac {35 i \cos (c+d x)}{192 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {35 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{128 a^3 d}
\]
[Out]
35/256*I*arctanh(1/2*sec(d*x+c)*a^(1/2)*2^(1/2)/(a+I*a*tan(d*x+c))^(1/2))/a^(5/2)/d*2^(1/2)+35/192*I*cos(d*x+c
)/a^2/d/(a+I*a*tan(d*x+c))^(1/2)-35/128*I*cos(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/a^3/d+1/6*I*cos(d*x+c)/d/(a+I*a*
tan(d*x+c))^(5/2)+7/48*I*cos(d*x+c)/a/d/(a+I*a*tan(d*x+c))^(3/2)
Rubi [A] (verified)
Time = 0.32 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3583, 3571, 3570, 212}
\[
\int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {35 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{128 \sqrt {2} a^{5/2} d}-\frac {35 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{128 a^3 d}+\frac {35 i \cos (c+d x)}{192 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {7 i \cos (c+d x)}{48 a d (a+i a \tan (c+d x))^{3/2}}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}
\]
[In]
Int[Cos[c + d*x]/(a + I*a*Tan[c + d*x])^(5/2),x]
[Out]
(((35*I)/128)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(5/2)*d) + ((I/
6)*Cos[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(5/2)) + (((7*I)/48)*Cos[c + d*x])/(a*d*(a + I*a*Tan[c + d*x])^(3/2
)) + (((35*I)/192)*Cos[c + d*x])/(a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (((35*I)/128)*Cos[c + d*x]*Sqrt[a + I*a*
Tan[c + d*x]])/(a^3*d)
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 3570
Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-2*(a/(b*f)), Subst[
Int[1/(2 - a*x^2), x], x, Sec[e + f*x]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 + b^
2, 0]
Rule 3571
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(d*
Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/(a*f*m)), x] + Dist[a/(2*d^2), Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan
[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && EqQ[m/2 + n, 0] && GtQ[n, 0]
Rule 3583
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(d*
Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/(b*f*(m + 2*n))), x] + Dist[Simplify[m + n]/(a*(m + 2*n)), Int[(d*Sec[
e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2*n]
Rubi steps \begin{align*}
\text {integral}& = \frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}+\frac {7 \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx}{12 a} \\ & = \frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}+\frac {7 i \cos (c+d x)}{48 a d (a+i a \tan (c+d x))^{3/2}}+\frac {35 \int \frac {\cos (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{96 a^2} \\ & = \frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}+\frac {7 i \cos (c+d x)}{48 a d (a+i a \tan (c+d x))^{3/2}}+\frac {35 i \cos (c+d x)}{192 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {35 \int \cos (c+d x) \sqrt {a+i a \tan (c+d x)} \, dx}{128 a^3} \\ & = \frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}+\frac {7 i \cos (c+d x)}{48 a d (a+i a \tan (c+d x))^{3/2}}+\frac {35 i \cos (c+d x)}{192 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {35 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{128 a^3 d}+\frac {35 \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{256 a^2} \\ & = \frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}+\frac {7 i \cos (c+d x)}{48 a d (a+i a \tan (c+d x))^{3/2}}+\frac {35 i \cos (c+d x)}{192 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {35 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{128 a^3 d}+\frac {(35 i) \text {Subst}\left (\int \frac {1}{2-a x^2} \, dx,x,\frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}}\right )}{128 a^2 d} \\ & = \frac {35 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{128 \sqrt {2} a^{5/2} d}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}+\frac {7 i \cos (c+d x)}{48 a d (a+i a \tan (c+d x))^{3/2}}+\frac {35 i \cos (c+d x)}{192 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {35 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{128 a^3 d} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.41 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.74
\[
\int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {i \sec ^3(c+d x) \left (-125-105 e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )-85 \cos (2 (c+d x))+40 \cos (4 (c+d x))+7 i \sin (2 (c+d x))+56 i \sin (4 (c+d x))\right )}{768 a^2 d (-i+\tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}}
\]
[In]
Integrate[Cos[c + d*x]/(a + I*a*Tan[c + d*x])^(5/2),x]
[Out]
((I/768)*Sec[c + d*x]^3*(-125 - 105*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I
)*(c + d*x))]] - 85*Cos[2*(c + d*x)] + 40*Cos[4*(c + d*x)] + (7*I)*Sin[2*(c + d*x)] + (56*I)*Sin[4*(c + d*x)])
)/(a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])
Maple [B] (verified)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf
count of optimal. 729 vs. \(2 (155 ) = 310\).
Time = 10.36 (sec) , antiderivative size = 730, normalized size of antiderivative =
3.80
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| method | result | size |
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| default |
\(-\frac {320 i \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+320 i \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-420 i \cos \left (d x +c \right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-448 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )-490 i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-210 i \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-448 \sin \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+420 \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sin \left (d x +c \right )-490 i \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+315 i \sec \left (d x +c \right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+210 \tan \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+210 \tan \left (d x +c \right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+105 i \left (\sec ^{2}\left (d x +c \right )\right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+210 \tan \left (d x +c \right ) \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-105 \tan \left (d x +c \right ) \sec \left (d x +c \right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )}{768 d \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+1\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (1+i \tan \left (d x +c \right )\right )^{2} a^{2}}\) |
\(730\) |
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[In]
int(cos(d*x+c)/(a+I*a*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
[Out]
-1/768/d/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)/(cos(d*x+c)+1)/(a*(1+I*tan(d*x+c)))^(1/2)/(1+I*tan(d*x+c))^2/a^2*(
320*I*cos(d*x+c)^2*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+320*I*cos(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-420*
I*cos(d*x+c)*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-448*(-c
os(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)*sin(d*x+c)-490*I*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-210*I*arctan(1/
2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-448*sin(d*x+c)*(-cos(d*x+c)/(
cos(d*x+c)+1))^(1/2)+420*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1
/2))*sin(d*x+c)-490*I*sec(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+315*I*sec(d*x+c)*arctan(1/2*(I*sin(d*x+c)-
cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+210*tan(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(
1/2)+210*tan(d*x+c)*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+
105*I*sec(d*x+c)^2*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+2
10*tan(d*x+c)*sec(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-105*tan(d*x+c)*sec(d*x+c)*arctan(1/2*(I*sin(d*x+c)
-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)))
Fricas [A] (verification not implemented)
none
Time = 0.25 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.51
\[
\int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {{\left (-105 i \, \sqrt {\frac {1}{2}} a^{3} d \sqrt {\frac {1}{a^{5} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {35 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{5} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{64 \, a^{2} d}\right ) + 105 i \, \sqrt {\frac {1}{2}} a^{3} d \sqrt {\frac {1}{a^{5} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {35 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{5} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{64 \, a^{2} d}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-48 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 39 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 125 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 46 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{768 \, a^{3} d}
\]
[In]
integrate(cos(d*x+c)/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="fricas")
[Out]
1/768*(-105*I*sqrt(1/2)*a^3*d*sqrt(1/(a^5*d^2))*e^(6*I*d*x + 6*I*c)*log(-35/64*(sqrt(2)*sqrt(1/2)*(I*a^2*d*e^(
2*I*d*x + 2*I*c) + I*a^2*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(1/(a^5*d^2)) - I)*e^(-I*d*x - I*c)/(a^2*d))
+ 105*I*sqrt(1/2)*a^3*d*sqrt(1/(a^5*d^2))*e^(6*I*d*x + 6*I*c)*log(-35/64*(sqrt(2)*sqrt(1/2)*(-I*a^2*d*e^(2*I*
d*x + 2*I*c) - I*a^2*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(1/(a^5*d^2)) - I)*e^(-I*d*x - I*c)/(a^2*d)) + s
qrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-48*I*e^(8*I*d*x + 8*I*c) + 39*I*e^(6*I*d*x + 6*I*c) + 125*I*e^(4*I*
d*x + 4*I*c) + 46*I*e^(2*I*d*x + 2*I*c) + 8*I))*e^(-6*I*d*x - 6*I*c)/(a^3*d)
Sympy [F]
\[
\int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {\cos {\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx
\]
[In]
integrate(cos(d*x+c)/(a+I*a*tan(d*x+c))**(5/2),x)
[Out]
Integral(cos(c + d*x)/(I*a*(tan(c + d*x) - I))**(5/2), x)
Maxima [B] (verification not implemented)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf
count of optimal. 2297 vs. \(2 (145) = 290\).
Time = 0.51 (sec) , antiderivative size = 2297, normalized size of antiderivative = 11.96
\[
\int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="maxima")
[Out]
1/3072*(544*(cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*
d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)^(3/4)*((-I*sqrt(2)*cos(6*d*x + 6*
c) - sqrt(2)*sin(6*d*x + 6*c))*cos(3/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*a
rctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)) + (sqrt(2)*cos(6*d*x + 6*c) - I*sqrt(2)*sin(6*d*x + 6*c))*si
n(3/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*
x + 6*c))) + 1)))*sqrt(a) + 12*(cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6
*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)^(1/4)*(29*((I*
sqrt(2)*cos(6*d*x + 6*c) + sqrt(2)*sin(6*d*x + 6*c))*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 +
(I*sqrt(2)*cos(6*d*x + 6*c) + sqrt(2)*sin(6*d*x + 6*c))*sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2
+ 2*(I*sqrt(2)*cos(6*d*x + 6*c) + sqrt(2)*sin(6*d*x + 6*c))*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c
))) + I*sqrt(2)*cos(6*d*x + 6*c) + sqrt(2)*sin(6*d*x + 6*c))*cos(5/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c),
cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)) + (19*I*sqrt(2)*cos(6*d*x + 6*
c) + 19*sqrt(2)*sin(6*d*x + 6*c) - 16*I*sqrt(2))*cos(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x +
6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)) - 29*((sqrt(2)*cos(6*d*x + 6*c) - I*sqrt(2
)*sin(6*d*x + 6*c))*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + (sqrt(2)*cos(6*d*x + 6*c) - I*sqr
t(2)*sin(6*d*x + 6*c))*sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*(sqrt(2)*cos(6*d*x + 6*c) -
I*sqrt(2)*sin(6*d*x + 6*c))*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + sqrt(2)*cos(6*d*x + 6*c) -
I*sqrt(2)*sin(6*d*x + 6*c))*sin(5/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arct
an2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)) - (19*sqrt(2)*cos(6*d*x + 6*c) - 19*I*sqrt(2)*sin(6*d*x + 6*c)
- 16*sqrt(2))*sin(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x
+ 6*c), cos(6*d*x + 6*c))) + 1)))*sqrt(a) - 105*(2*sqrt(2)*arctan2((cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*
x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), co
s(6*d*x + 6*c))) + 1)^(1/4)*sin(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arct
an2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)), (cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(
1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) +
1)^(1/4)*cos(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c
), cos(6*d*x + 6*c))) + 1)) + 1) - 2*sqrt(2)*arctan2((cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 +
sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)
)) + 1)^(1/4)*sin(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x
+ 6*c), cos(6*d*x + 6*c))) + 1)), (cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(si
n(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)^(1/4)*cos(1
/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x +
6*c))) + 1)) - 1) - I*sqrt(2)*log(sqrt(cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arcta
n2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)*cos(1/
2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x +
6*c))) + 1))^2 + sqrt(cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c
), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)*sin(1/2*arctan2(sin(1/3*
arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1))^2 + 2
*(cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))
)^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/3*arctan2(sin(6*
d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)) + 1) + I*sqrt(2)*lo
g(sqrt(cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x +
6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)*cos(1/2*arctan2(sin(1/3*arctan2(sin(6*d
*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1))^2 + sqrt(cos(1/3*arc
tan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1
/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)*sin(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d
*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1))^2 - 2*(cos(1/3*arctan2(sin(6*d*x + 6*c
), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x
+ 6*c), cos(6*d*x + 6*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), c
os(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)) + 1))*sqrt(a))/(a^3*d)
Giac [F]
\[
\int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate(cos(d*x+c)/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="giac")
[Out]
integrate(cos(d*x + c)/(I*a*tan(d*x + c) + a)^(5/2), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x
\]
[In]
int(cos(c + d*x)/(a + a*tan(c + d*x)*1i)^(5/2),x)
[Out]
int(cos(c + d*x)/(a + a*tan(c + d*x)*1i)^(5/2), x)